Tabloids first buzzed about a possible new lady love in the Oscar winner's life when reports surfaced that the two enjoyed each other's company at Clooney's Lake Como villa in early July.

"I'm in heaven," Keibler Tweeted July 8. The gushing continued: "I am the luckiest girl in the world," she added the next day.

But follow-up Tweets suggested the reported Italian getaway wasn't so one-on-one. "Our whole villa, 19 of us, are on a day trip," she Tweeted July 13. She added, "You can't believe anything you read and only half of what you see!"

A source tells PEOPLE that the two "have run into each other this summer." But another source adds that they have only been "spending time together" over the last few weeks and are having "fun" – not dating. "It's not just a friendship at this point," the source says.

Clooney has been promoting his upcoming political drama, The Ides of March, and otherwise staying off the radar.